Laminar Couette Flow with Imposed Pressure Gradient

In this application, AcuSolve is used to simulate the viscous flow of water between a moving and a stationary plate with an imposed pressure gradient. AcuSolve results are compared with analytical results described in White (1991). The close agreement of AcuSolve results with analytical results validates the ability of AcuSolve to model cases with imposed pressure gradients.

Problem Description

The problem consists of air between two plates in a two dimensional domain, as shown in the following image, which is not drawn to scale. The domain is 1.0 m high and 1.5 m long. The top plate moves with a constant velocity of 3.0 m/sec and the bottom plate is fixed. There is a mean-pressure gradient of -12 Pa/m applied to the bulk fluid in the streamwise direction. The problem is simulated with periodic boundaries in the streamwise direction. The induced flow field is laminar and exhibits a steady state behavior. The flow field develops from the pressure gradient, the motion of the top plate, and the viscous shear stresses near the plates.
Figure 1. Critical Dimensions and Parameters for Simulating Laminar Couette Flow with an Imposed Pressure Gradient


The simulation was performed as a two dimensional problem by restricting flow in the out-of-plane direction through the use of a mesh that is one element thick.
Figure 2. Mesh used for Simulating Laminar Couette Flow with an Imposed Pressure Gradient


AcuSolve Results

The AcuSolve solution converged to a steady state and the results reflect the mean flow conditions. The greatest velocity is located at approximately 40 percent of the channel height, closer to the moving plate. The flow develops as a result of the pressure gradient and the shear stress acting on the fluid near both the moving plate and the stationary plate.
Figure 3. Z-Velocity Contours and Velocity Vectors


Figure 4. Z Velocity Plotted Against Height Above the Bottom of the Flow Field (Z Velocity is Presented on the X Axis to Better Represent the Velocity Profile in the Direction of Flow)


Summary

The velocity profile computed by AcuSolve agrees well with the analytical solution for this application. The velocity profile arises due to the combination of the imposed pressure gradient and the constant upper-wall velocity.
Note: The combination of these effects results in the asymmetric velocity profile that is reflected in the results.

Simulation Settings for Laminar Couette Flow with Imposed Pressure Gradient

SimLab database file: <your working directory>\couette_flow\couette_flow.slb

Global

  • Problem Description
    • Solution Type - Steady State
    • Flow - Laminar
  • Auto Solution Strategy
    • Relaxation factor - 0.2
  • Material Model
    • Air
      • Density - 1.0 kg/m3
      • Viscosity - 1.0 kg/m-sec
  • Body Force
    • DP/DL
      • Gravity
        • Z-component - 18.0 m/sec2

    Model

  • Volumes
    • Fluid
      • Element set
        • Material model - Air
        • Body force - DP/DL
  • Surfaces
    • Max_X
      • Simple Boundary Condition
        • Type - Symmetry
    • Max_Y
      • Simple Boundary Condition
        • Type - Wall
        • Wall velocity type - Cartesian
        • Z-velocity - 3.0 m/s
    • Max_Z
      • Simple Boundary Condition - (disabled to allow for periodic conditions to be set)
    • Min_X
      • Simple Boundary Condition
        • Type - Symmetry
    • Min_Y
      • Simple Boundary Condition
        • Type - Wall
    • Min_Z
      • Simple Boundary Condition - (disabled to allow for periodic conditions to be set)
  • Periodics
    • Periodic 1
      • Periodic Boundary Conditions
        • Type - Periodic

References

F. M. White. “Viscous Fluid Flow”. Section 3-2.3. McGraw-Hill Book Co., Inc. New York. 1991.